# 1.6 Graph Transformations

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1.6 Graph Transformations
Shelby Sell Sammie Meddaugh Emily Wojahn

INTRODUCTION

Vocabulary Transformation: Functions that map real number to real numbers. Rigid Transformations: Leave the side and shape of the graph unchanged (horizontal and vertical translations, reflections). Non Rigid Transformations: Distort the shape of a graph (horizontal or vertical stretches and shapes).

Translations If c is a positive real number:
Horizontal y= f(x-c) A translation to the right by c units y= f(x+c) A translation to the right by c units Vertical y= f(x) + c A translation up by c units y= f(x) - c A translation down by c units

Examples y= abs(x) y= abs(x – 2) 2 y= x2 y= x2 + 3

Reflections Across the x-axis (x,y) (x,-y) y= -f(x) Across the y-axis (x,y) (-x,y) y= f(-x) Through the origin (x,y) (-x.-y) y= -f(-x)

Examples Reflection over y= x axis Reflection over y axis
Reflection over x axis

Stretches/Shrinks y= f (x/c) y= c f(x)
Horizontal Stretch/Shrink A stretch by a factor of c if c>1 A shrink by a factor of c if c<1 Vertical Stretch/Shrink A stretch by a factor of c if c> 1 y= f (x/c) y= c f(x)

Examples y = x² Vertical Stretches Graph y = 3x² Graph of y = x²
Multiply all red values by 3 to get coordinates for the new graph. "Transformations on the Basic Parabola." W.A.E.C.E. Math Help. N.p., n.d. Web.

Solution to y = 3x² Graph of y = x² with a stretch of 3.

Combining Transformations in Order
1.) Horizontal shift 2 units to the right y=(x-2) ² 2.) Stretch Vertically by factor 3 y=3(x-2) ² 3.) Vertical Translation 5 units up y=3(x-2) ² +5 Given y=x²

1.) Horizontal shift 2 units to the right y=(x-2) ²

2.) Stretch Vertically by factor 3 y=3(x-2) ²

3.) Vertical Translation 5 units up y=3(x-2) ² +5

Absolute Value- Distance
Y= f(x) Entire functions absolute value (change negative y values to positive) Y= - f(x) Only Negative y values

More absolute value

1.) Assessment A) y = ½x2 + 2 B) y = 3x2 + 2 C) y = 3(x + 2)2
D) y = ½(x + 2)2

2.) C) y = 2(x - 3)2 A) y = ½(x - 3)2 D) y = 2(x + 3)2
B) y = ½(x + 3)2

3.) C y = 2(x + 1)4 A y = 0.5(x + 1)4 D y = 2(x - 1)4
B y = 0.5(x - 1)4

4. Describe how the graph of y= x² can be transformed to the graph of the given equation Y=x²-3 A) Vertical translation up 3 units B) Horizontal translation to the right 3 units C) Horizontal translation to the left 3 units D) Vertical translation down 3 units

5.) A) y=f(3x) C) y=f(9x/3) B) y=3f(x) D)y=f(x)/3
Given function f, which of the following represents a vertical stretch by a factor of 3. A) y=f(3x) B) y=3f(x) C) y=f(9x/3) D)y=f(x)/3

6.) Given a function f, which of the following represents a vertical translation of 2 units upward, followed by a reflection across the y-axis. A) y=f(-x) + 2 C) y= -f(x-2) B) y= 2-f(x) D) f(x) -2

7.) TRUE OR FALSE? The function y=f(x+3) represents a translation to the right by 3 units of the graph of y = f(x).

8.) TRUE OR FALSE? The function of y=f(x)-4 represents a translation down 4 units of the graph of y=f(x)

9.) Write an equation whose graph is
Y=x ²; a vertical stretch by a factor of 3, then shift right 4 units A) y=3(x-4) ² C) y=3x ² -4 B) y=-3x ² +4 D) y=3(x+4) ²

10.) Write an equation whose graph is
Y= x ; a shift left 2 units, then a vertical stretch y a factor of 2, and finally a shift down 4 units. A) 2 x C) 2 x-2 +4 B) 2(x+2) -4 D) 3(x-2) +4

Answers 1.) B 2.) A 3.) A 4.) D 5.) B 6.) A 7.) False, it is translated left. 8.)True 9.) A 10.)A

Sources Pre calculus- Eighth edition book "Transformations on the Basic Parabola." W.A.E.C.E. Math Help. N.p., n.d. Web.

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